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3 votes
If x+y+z=0 what is the value of

{x}^(3) + {y}^(3) + {z}^(3)

User Hoser
by
8.2k points

2 Answers

3 votes

I don't think that there's a unique answer, because it depends on the numbers you choose: for example, if you pick
x = -1,\ y=0,\ z= 1, then you have


x+y+z=0,\quad x^3+y^3+z^3 = 0

But for example, if you if you pick
x = -2,\ y=1,\ z= 1, then you have


x+y+z=0,\quad x^3+y^3+z^3 = -8+1+1 = -6

Even if you wanted to use formula, you can at most solve one variable in terms of the other two:


x+y+z \iff x=-y-z \iff x^3 = -y^3 - 3 y^2 z - 3 y z^2 - z^3

and thus


x^3+y^3+z^3 = -y^3 - 3 y^2 z - 3 y z^2 - z^3+y^3+z^3 = - 3 y^2 z - 3 y z^2

User Mykoman
by
7.6k points
2 votes


x {}^(3) + y {}^(3) + z {}^(3) = (x + y + z)(x {}^(2) + y {}^(2) + z {}^(2) - xy - yz - xz) + 3xyz
if we put the value of x+y+z in the equation so the 3xyz will remain so the answer is 3xyz.
User Cronos
by
8.0k points

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